Adaptive Piecewise–Affine Inverse Modeling of Hybrid Dynamical Systems

Elhamifar, Ehsan; Burden, Samuel A.; Sastry, S. Shankar

doi:10.3182/20140824-6-za-1003.02610

Cited by 4 publications

(4 citation statements)

References 21 publications

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“…Mixed integer formulations that mainly focus on hinging hyperplanes and piecewise affine Wiener models have been proposed (Roll et al, 2004) but as the number of integer variables scale with number of data points such approaches are only applicable in small data regime. On the contrary, researchers have also focused on convex formulations where first they estimate a set of submodels and then select few of them that explains the data (Elhamifar et al, 2014), but the approach relies on restricting the parameter space and can be overly conservative.…”

Section: Related Workmentioning

confidence: 99%

Learning Linear Complementarity Systems

Jin¹,

Aydinoglu²,

Halm³

et al. 2021

Preprint

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This paper investigates the learning, or system identification, of a class of piecewiseaffine dynamical systems known as linear complementarity systems (LCSs). We propose a violation-based loss which enables efficient learning of the LCS parameterization, without prior knowledge of the hybrid mode boundaries, using gradient-based methods. The proposed violation-based loss incorporates both dynamics prediction loss and a novel complementarity -violation loss. We show several properties attained by this loss formulation, including its differentiability, the efficient computation of first-and second-order derivatives, and its relationship to the traditional prediction loss, which strictly enforces complementarity. We apply this violation-based loss formulation to learn LCSs with tens of thousands of (potentially stiff) hybrid modes. The results demonstrate a state-of-the-art ability to identify piecewise-affine dynamics, outperforming methods which must differentiate through non-smooth linear complementarity problems.

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Section: Related Workmentioning

confidence: 99%

Learning Linear Complementarity Systems

Jin¹,

Aydinoglu²,

Halm³

et al. 2021

Preprint

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“…-Minimizing the Lagrangian function in (13) with respect to C subject to the probability simplex constraints {1 C = 1 , C ≥ 0} can be done using the algorithm in [57] with O(M log(M )N ) computational time (O(M N ) expected time using the randomized algorithm in [57]). Notice that we can solve (13) via N independent smaller optimization programs over the N columns of C. Thus, having P parallel processing resources, we can reduce the computational time to O(M log(M ) N/P ) (or O(M N/P ) expected time using the randomized algorithm in [57]). As a result, the proposed ADMM implementation of our algorithm can be performed in O(M log(M )N ) computational time, while we can reduce the computational time to O( M N/P log(M )) using P parallel resources.…”

Section: Dsimplementationmentioning

confidence: 99%

“…F INDING a subset of a large number of models or data points, which preserves the characteristics of the entire set, is an important problem in machine learning and data analysis with applications in computer vision [1], [2], [3], [4], [5], [6], image and natural language processing [7], [8], bio/health informatics [9], [10], recommender systems [11], [12] and more [13], [14], [15], [16]. Such informative elements are referred to as representatives or exemplars.…”

Section: Introductionmentioning

confidence: 99%

“…Data representatives help to summarize and visualize datasets of text/web documents, images and videos (see Figure 1), hence, increase the interpretability of large-scale datasets for data analysts and domain experts [1], [2], [3], [9], [17]. Model representatives help to efficiently describe complex phenomena or events using a small number of models or can be used for model compression in ensemble models [13], [18]. More importantly, the computational time and memory requirements of learning and inference algorithms, such as the Nearest Neighbor (NN) classifier, improve by working on representatives, which contain much of the information of the original set [15].…”

Section: Introductionmentioning

confidence: 99%

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Dissimilarity-Based Sparse Subset Selection

Elhamifar

Sapiro

Sastry

2016

IEEE Trans. Pattern Anal. Mach. Intell.

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152

102

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Finding an informative subset of a large collection of data points or models is at the center of many problems in computer vision, recommender systems, bio/health informatics as well as image and natural language processing. Given pairwise dissimilarities between the elements of a 'source set' and a 'target set,' we consider the problem of finding a subset of the source set, called representatives or exemplars, that can efficiently describe the target set. We formulate the problem as a row-sparsity regularized trace minimization problem. Since the proposed formulation is, in general, NP-hard, we consider a convex relaxation. The solution of our optimization finds representatives and the assignment of each element of the target set to each representative, hence, obtaining a clustering. We analyze the solution of our proposed optimization as a function of the regularization parameter. We show that when the two sets jointly partition into multiple groups, our algorithm finds representatives from all groups and reveals clustering of the sets. In addition, we show that the proposed framework can effectively deal with outliers. Our algorithm works with arbitrary dissimilarities, which can be asymmetric or violate the triangle inequality. To efficiently implement our algorithm, we consider an Alternating Direction Method of Multipliers (ADMM) framework, which results in quadratic complexity in the problem size. We show that the ADMM implementation allows to parallelize the algorithm, hence further reducing the computational time. Finally, by experiments on real-world datasets, we show that our proposed algorithm improves the state of the art on the two problems of scene categorization using representative images and time-series modeling and segmentation using representative models.

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An iterative method and its application to stable inversion

et al. 2018

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In this paper, we study convergence and data dependence of SP and normal-S iterative methods for the class of almost contraction mappings under some mild conditions. The validity of these theoretical results is confirmed with numerical examples. It has been observed that a special case of SP iterative method, namely, normal-S iterative method, performs better and so the latter is implemented in the stable inversion of nonlinear discrete time dynamical systems to yield convergence results when Picard iterative method diverges. This is also illustrated with a numerical example. Our work extends and improves upon many results existing in the literature.

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Adaptive Piecewise–Affine Inverse Modeling of Hybrid Dynamical Systems

Cited by 4 publications

References 21 publications

Learning Linear Complementarity Systems

Learning Linear Complementarity Systems

Dissimilarity-Based Sparse Subset Selection

An iterative method and its application to stable inversion

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